Running Lanes and Extra Steps
Ivars Peterson
When going out to your local running track for a workout, you sometimes find
that you are allowed to use only certain lanes for training. On a standard
quadrant track, however, the outer lanes are longer than the inner lanes. That
presents a problem for someone using the track for speed workouts.
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A four-lane equal-quadrant running track.
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With that in mind, a runner once submitted the following question to
Runner's World magazine: How can I find the exact distances of all the
lanes?
"This is a common question," veteran runner Elliott A. Weinstein of Baltimore
wrote in the April Mathematics Magazine. ". . . the inner lanes are
yielded to faster runners by protocol, to slower runners and walkers who ignore
or are ignorant of the protocol, and to the entire high school marching band,
which just happens to be practicing on the track during your workout and is not
bound by the protocol."
Two coaches replied to the Runner's World question. One coach had gone
out to his 400-meter, eight-lane college track and measured the distances in
lanes 4 and 8 (but he didn't say how). Given these two values, he suggested
determining the distances of the remaining lanes by interpolation. The other
coach recommended running a lap in each lane while holding the pace steady
throughout and noting the time differences.
Weinstein suggested that a mathematical approach would give a much better
answer.
A typical 400-meter quadrant track consists of two parallel straightaways
connected at the ends by concentric semicircles. For any closed, convex curve,
such as a lane divider, the extra distance traveled along a path that is
everywhere a distance d away from and outside the curve is simply
2?d.
"Note that this result is independent of the distance around the interior
curve," Weinstein remarked.
For any given track, every lane has the same width. Hence, for a track with
lane width w, the extra distance around the track when running in lane
n is 2?w(n ? 1).
The standard lane width for most high school and college outdoor tracks in
the United States is 42 inches. This gives 6.70 meters of extra distance per
lane per lap. So lane 4 would have a distance of 420 meters and lane 8 a
distance of 447 meters, rounded to the nearest meter.
It turns out that the coach's measurements were off by 1 meter in one case
and 8 meters in the other. He might have obtained better results simply by
taking the differences in the lanes' staggered starting marks for an appropriate
track event.
Nonetheless, Weinstein concludes, "it's safer and easier just to do the
math!"
References:
Weinstein, E.A. 2003. Math bite: The extra distance in an
outer lane of a running track. Mathematics Magazine
76(April):149-150.
The United States Tennis Court & Track Builders
Association provides guidelines for constructing running tracks at http://www.ustctba.com/guidelines-track/contents.html.
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A collection of Ivars Peterson's early
MathTrek articles, updated and illustrated, is now available as the
Mathematical Association of America (MAA) book Mathematical Treks: From
Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.
| Comments are welcome. Please send messages to Ivars
Peterson at ip@sciencenews.org.
 Ivars Peterson is the mathematics writer and online editor at
Science News. He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, Fatal Defect, The Jungles of Randomness, and Fragments of Infinity. He also writes for the
children's magazine Muse (see MatheMUSEments at http://home.att.net/~mathtrek/). The
Mathematical Association of America has published a collection of his online
MathTrek
articles.
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